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6. Three forces are represented in magnitude, direction, and line of action by the sides of a triangle taken in order; prove that their resultant is a couple the torque of which equals, numerically, twice the area of the triangle.
6. Three forces act along the sides of an equilateral triangle; find the condition which will make their resultant pass through the center of the triangle.
57. Friction on Journal Bearings.  If the horizontal shaft of Fig. 38 fits perfectly in its bearings the friction which comes into play is a sliding friction, therefore the laws of sliding friction may be assumed to hold good. The most important of these laws is : the frictional force which comes into play is proportional to the normal reaction, that is, in the relation
AC is independent of N. We will assume therefore that this law holds at each point of the surface of contact and thus reduce the problem under discussion to one of sliding friction. There is an important difference, however, between the problem under discussion and the problems on friction which we have already discussed. In the present problem the normal reaction is not the same at all the points of the surfaces in contact. We must apply, therefore, the laws of friction to small elements of surfaces of contact over which the normal reaction may be considered to be constant. Let the element of surface be a strip, along the length of the shaft, which subtends an angle dd at the axis of the shaft. Further let dN be the normal reaction over this element of surface, and df be the corresponding frictional force; then we have
= fj.p  I  a do,
where p is the normal reaction per unit area or the pressure, a is the radius of the shaft, and I the length of the bearing.